Planar metamaterials for control of electromagnetic wave guidance and radiation

ABSTRACT

A linear metamaterial comprises a transmission line, having a linear dimension, and is loaded with capacitors, and shunted with an inductor such that for an electromagnetic wave, having a wavelength greater than the linear dimension and traveling along the axis of the transmission line, the effective permeability and permittivity of the metamaterial are simultaneously negative. Applications for the metameterial are also disclosed.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application claims the benefit of U.S. ProvisionalPatent Application No. 60/384,080 filed on May 31, 2002, which is herebyincorporated by reference herein.

FIELD OF THE INVENTION

[0002] The present invention relates generally to the control andguidance of electromagnetic radiation and in particular to a left-handedmetamaterial for controlling and guiding electromagnetic radiation andto applications therefor.

BACKGROUND OF THE INVENTION

[0003] The concept of a negative index of refraction, originallyproposed by Veselago in the 1960s, suggested the possibility ofmaterials in which the permittivity and permeability could be madesimultaneously negative. Veselago termed these left-handed media (LHM),because the vectors E, H, and k would form a left-handed triplet insteadof a right-handed triplet, as is the case in conventional, right-handedmedia (RHM). In such a material the phase velocity and Poynting vectorare antiparallel. Recently, novel 3-dimensional (3-D) electromagneticmaterials have successfully demonstrated negative refraction bysynthesizing a negative refractive index. These artificial dielectrics(metamaterials) consist of loosely coupled unit cells composed of thinwire strips and split-ring resonators to synthesize negativepermittivity and permeability, respectively. In these metamaterials, thechoice of operating frequency is restricted to the region of theresonance, which results in a highly dispersive, narrow and behaviourwith strong associated absorption losses.

[0004] Other structures with magnetic properties to refractelectromagnetic radiation at negative angles have also been considered.For example, International PCT Application No. WO 00/41270 discloses astructure that exhibits magnetic properties when it receives incidentelectromagnetic radiation. The structure is formed from an array ofcapacitive elements, each of which is smaller, and preferably muchsmaller, than the wavelength of the incident electromagnetic radiation.Each capacitive element has a low-resistance conducting path associatedwith it and is such that a magnetic component of the incidentelectromagnetic radiation induces an electrical current to flow around apath and through the associated capacitive element. The creation of theinternal magnetic fields generated by the flow of the induced electricalcurrent gives rise to the structure's magnetic properties.

[0005] International PCT Application No. WO 02/03500 discloses amicrostructured magnetic material having a magnetic permeability ofnegative value but unity magnitude over a particular radio frequencyrange.

[0006] Although the above references disclose structures with magneticproperties to refract electromagnetic radiation at negative angles,improved materials exhibiting negative refractive indices are desired.

[0007] It is therefore an object of the present invention to provide anovel left-handed metamaterial for controlling and guidingelectromagnetic radiation and novel applications therefor.

SUMMARY OF THE INVENTION

[0008] According to one aspect of the present invention there isprovided a planar metamaterial comprising:

[0009] two substantially orthogonal, coplanar sets of transmissionlines, said transmission lines being spaced with a periodicity, loadedwith capacitors with said periodicity, and shunted with inductors withsaid periodicity such that for an electromagnetic wave, having awavelength greater than said periodicity and traveling along the planeof said transmission lines, the effective permeability and permittivityof said metamaterial are simultaneously negative.

[0010] According to another aspect of the present invention there isprovided a linear metamaterial comprising:

[0011] a transmission line, having a linear dimension, and being loadedwith capacitors, and shunted with an inductor such that for anelectromagnetic wave, having a wavelength greater than said lineardimension and traveling along the axis of said transmission line, theeffective permeability and permittivity of said metamaterial aresimultaneously negative.

[0012] According to yet another aspect of the present invention there isprovided a planar resonance cone metamaterial comprising:

[0013] a first set of transmission lines, spaced with a periodicity, andloaded with capacitors with said periodicity;

[0014] a second set of transmission lines, substantially orthogonal andcoplanar with said first set of transmission lines, said second set oftransmission lines being spaced with said periodicity, and loaded withinductors with said periodicity, said first and second sets oftransmission lines exhibiting characteristics such that for anelectromagnetic wave, having a wavelength greater than said periodicityby an order of magnitude, and traveling along the linear axis of saidfirst set of transmission lines, the effective permittivity of saidmetamaterial is positive such that for an electromagnetic wave, having awavelength greater than said periodicity by an order of magnitude andtraveling along the linear axis of said second set of transmissionlines, the effective permittivity of said metamaterial is negative.

[0015] According to yet another aspect of the present invention there isprovided a planar resonance cone metamaterial comprising:

[0016] a first set of transmission lines, spaced with a firstperiodicity, and loaded with capacitors with a second periodicity;

[0017] a second set of transmission lines, substantially orthogonal andcoplanar with said first set of transmission lines, said second set oftransmission lines being spaced with said second periodicity, and beingloaded with inductors with said first periodicity whereby for anelectromagnetic wave, having a wavelength greater than said first andsecond periodicities by an order of magnitude and traveling along thelongitudinal axis of said first set of transmission lines, the effectivepermittivity of said metamaterial is positive and for an electromagneticwave having a wavelength greater than said first and second priodocitiesand traveling along the longitudinal axis of said second set oftransmission lines, the effective permittivity of said metamaterial isnegative.

[0018] According to yet another aspect of the present invention there isprovided a near field focusing device comprising:

[0019] a first set of transmission lines, said first set of transmissionlines being spaced with a first periodicity and loaded with capacitorswith a second periodicity;

[0020] a second set of transmission lines, substantially orthogonal to,and coplanar with said first set of transmission lines, said second setof transmission lines being spaced with said second periodicity, loadedwith capacitors with said first periodicity, and shunted with inductors,said first set of transmission lines intersecting said second set oftransmission lines such that for an electromagnetic wave, having awavelength greater than said first or second periodicity by an order ofmagnitude, and traveling along the plane of said transmission lines, theeffective permeability and permittivity of said metamaterial aresimultaneously negative; and

[0021] a planar waveguide, having a flat extent, coupled to saidtransmission lines, such that said flat extent is parallel to one set oftransmission lines.

[0022] According to yet another aspect of the present invention there isprovided a phase-shifting device comprising:

[0023] a transmission line, having a linear dimension and characteristicimpedance;

[0024] capacitors loaded on said transmission line; and

[0025] an inductor shunting said transmission line, said transmissionline exhibiting characteristics such that for an electromagnetic wave,having a wavelength greater than said linear dimension by an order ofmagnitude and traveling along the axis of said transmission line, theeffective permeability and permittivity of transmission aresimultaneously negative and such that said transmission line provides acontinuous passband for said electromagnetic wave along saidtransmission line.

[0026] According to yet another aspect of the present invention there isprovided a near field focussing device comprising:

[0027] a parallel-plate waveguide; and

[0028] a negative refractive index metamaterial in line with saidwaveguide and forming an interface therewith.

[0029] According to yet another aspect of the present invention there isprovided a coupled-line coupler comprising:

[0030] a microstrip line; and

[0031] a left-handed material coupled to an edge of said microstripline.

[0032] The present invention provides advantages in that themetamaterial is capable of internal positive and negative waverefraction, and guided wave beam formation, steering, and focusing overregions that are smaller than the wavelength of incident electromagneticradiation. Moreover, the beam steering capability can be extended beyondthe physical boundaries of the metamaterial into the surrounding spaceto produce a controllable radiation pattern.

BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Embodiments of the present invention will now be described morefully with reference to the accompanying drawings in which:

[0034]FIG. 1 shows phase matching at an interface between a right-handedmaterial (RHM) and a generic material i.e. a right-handed material or aleft-handed material;

[0035]FIG. 2 is a two-dimensional (2-D) loaded left-handed transmissionline unit cell;

[0036]FIG. 3 shows a plane wave illuminating a RHM/LHM interface at 2GHz;

[0037]FIG. 4 illustrates the focusing effect at an interface betweenright-handed and left-handed media;

[0038]FIG. 5 shows a point source illuminating a RHM/RHM interface at 2GHz;

[0039]FIG. 6 shows a point source illuminating a RHM/LHM interface at 2GHz;

[0040]FIG. 7 shows a focusing device including a negative refractiveindex metamaterial interfaced with a parallel-plate waveguide at 1.5GHz;

[0041]FIG. 8 shows experimental data demonstrating focal regionsobserved in the focusing device of FIG. 7 at 1.55 GHz and 1.65 GHz andthe lack thereof at 2.55 GHz when operating beyond the left-handedfrequency band;

[0042]FIG. 9 shows a planar resonance cone metamaterial including asquare-celled transmission-line network containing inductors andcapacitors oriented orthogonally and positioned over a ground plane;

[0043]FIG. 10 shows one side of the square-celled transmission-linenetwork of FIG. 9 illustrating attached coaxial connectors;

[0044]FIG. 11 shows an opposite side of the square-celledtransmission-line network of FIG. 9, with chip capacitors andmeander-line inductors in place.

[0045]FIG. 12 shows single-stage, 2-stage, 4-stage and 8-stage 0°phase-shifting circuits;

[0046]FIG. 13 is a schematic diagram of a 1-D phase-shifting unit cell;

[0047]FIG. 14 shows an unmatched dispersion relation for thephase-shifting unit cell of FIG. 13;

[0048]FIG. 15 shows a matched dispersion relation for the phase-shiftingunit cell of FIG. 13;

[0049]FIG. 16 shows phase responses of a +10° phase-shifting device, a−350° transmission line, a −10° phase-shifting device and +10° L-C line;

[0050]FIG. 17 shows phase responses of one-stage, 4-stage and 8-stage 0°phase-shifting devices;

[0051]FIG. 18 shows phase and magnitude responses of +10° and −10°4-stage phase-shifting devices;

[0052]FIG. 19 shows a 4-stage +10° phase-shifting device;

[0053]FIG. 20 is a plasma parameter diagram showing elliptic andhyperbolic regions;

[0054]FIG. 21 is a schematic diagram of a focusing device in the form ofa uniform anisotropic planar L-C grid over ground, with corner feed andresistive edge-loading;

[0055]FIG. 22 shows a uniform-grid moment-method simulation displayingcorner-fed resonance cones at eight frequencies;

[0056]FIG. 23 shows node voltage across 50Ω terminating resistors atnumbered points around the edge of the grid of FIG. 21, derived from thesimulation data displayed in FIG. 22;

[0057]FIG. 24 shows a focusing device in the form of a non-uniform L-Cgrid, wherein the inductors and capacitors have been transposed in theupper part of the grid and wherein the transition region includes a rowof elongated cells;

[0058]FIG. 25 shows the physical layout of the grid of FIG. 24;

[0059]FIG. 26 shows resonance cone refraction at 1.2 GHz;

[0060]FIG. 27 shows a focusing simulation at 1.3 GHz showing thegrid-to-ground voltage magnitude;

[0061]FIG. 28 shows focusing measurements at 1.3 GHz showing thenormalized S21 magnitude;

[0062]FIG. 29 shows a focusing simulation at 1.7 GHz showing thegrid-to-ground voltage magnitude;

[0063]FIG. 30 shows focusing measurements at 1.7 GHz showing thenormalized S21 magnitude;

[0064]FIG. 31 shows a MS/LHM coupled-line coupler and schematic of acoupler unit cell;

[0065]FIG. 32 shows a dispersion diagram for MS and LHM lines used inthe coupler of FIG. 31 and photograph of a MS/LHM coupler and aconventional MS/MS coupler of the same length, line spacing andpropagation constant;

[0066]FIG. 33 shows a spatial Fourier transform of line voltages on theMS line and LHM line of the MS/LHM coupled-line coupler of FIG. 31;

[0067]FIG. 34 shows the coherence length of the MS/LHM coupled-linecoupler of FIG. 31;

[0068]FIG. 35 shows a comparison of a MS/MS coupler with a MS/LHMcoupler of the same length, line spacing and propagation constant forcoupled power, isolation, reflection, and arbitrary coupling;

[0069]FIG. 36 shows MS/LHM branch-line couplers;

[0070]FIG. 37 shows even/odd mode equivalent circuits of a type 1 MS/LHMbranch-line coupler;

[0071]FIGS. 38a to 38d show scattering parameters of type 1 MS/LHMbranch-line couplers compared to a regular branch-line coupler andscattering parameters of a type 2 MS/LHM branch-line coupler;

[0072]FIG. 39 shows a planar transmission-line left-handed lens;

[0073]FIG. 40 shows the measured vertical electric field detected 0.8 mmabove the surface of the lens of FIG. 39 at 1.057 GHz;

[0074]FIG. 41 shows the measured vertical electric field above row 0 ofthe lens of FIG. 39 at 1.057 GHz;

[0075]FIG. 42 shows the measured vertical electric field at the source(dashed curve) and image (solid curve) planes along with thediffraction-limited image (solid curve with circles);

[0076]FIG. 43 shows an antenna formed of 6 macrocells of resonance conemetamaterial; and

[0077]FIGS. 44 and 45 show the grid-to-ground voltage of a six cellantenna operating at 1 GH_(z) and the resulting radiation pattern.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0078] The present invention relates generally to a metamaterial thatsupports propagating backward waves and thus exhibits left-handedness.Such metamaterials inherently support 2-D wave propagation, which isdesirable for antennas, antenna beam formers, planar spectrum analyzers,compact RF/microwave lenses and antennas, phase compensators,antenna-integrated multiplexers and other microwave circuitapplications.

[0079] To illustrate the concept of negative refraction, consider aphase-matching argument at the interface between a right-handed mediumM1 and another generic medium M2, as shown in FIG. 1. The sign of theindex of refraction of medium M2 is not a concern. Consider an incidentplane wave in medium M1 with a wave vector k₁ (i.e. such that thex-component of k₁ is positive). A refracted wave with a wave vector k₂is then established in medium M2 such that the tangential wave vectorcomponents k_(1t) and k_(2t) are equal across the interface. This is thebasis for Snell's Law, and it permits two scenarios for the orientationof k₂, represented as Case 1 and Case 2 in FIG. 1. The conservation ofenergy also insists that the normal components of the Poynting vectorsS₁ and S₂ remain in the positive-x direction through both media. If themedium M2 is a conventional RHM, then refraction occurs as illustratedby Case 1. However, if medium M2 is a medium supporting propagatingbackward waves (i.e. a LHM), it is implied that power is propagatedalong the direction of phase advance, which requires in FIG. 1 that k₂and S₂ be antiparallel. Consequently, the direction of k₂ is specifieduniquely for backward-wave structures as illustrated by Case 2. Undersuch conditions, power is refracted through an effectively negativeangle, which implies a negative index of refraction.

[0080] Metamaterials exhibiting negative indices of refractions andapplications therefor that enable electromagnetic radiation to becontrolled and guided will now be described.

[0081] Planar Metamaterials

[0082] Turning now to FIG. 2, a shunt-centered unit cell for ametamaterial or left-handed medium (LHM) that synthezises a negativerefractive index is shown. In this particular example, the metamaterialis in the form of a two-dimensional (2-D), L-C loaded substantiallyorthogonal, coplanar transmission line network. In the unit cell,Z_(C)=(jωC)⁻¹ and Y_(L)=(jωL)⁻¹ represent the per-unit-length capacitorimpedance and inductor admittance, respectively.

[0083] The unit cell dimensionality represented by Δd is provided by thehost transmission line medium. At the continuous medium limit, Δd<<λ,the 2-D telegrapher's equations representing the structure of the unitcell can be expressed as: $\begin{matrix}{{{\frac{\partial v_{y}}{\partial z} = {- {i_{z}( \frac{1}{j\quad \omega \quad C\quad \Delta \quad d} )}}},{\frac{\partial v_{y}}{\partial x} = {- {i_{x}( \frac{1}{j\quad \omega \quad C\quad \Delta \quad d} )}}}}{and}} & (1) \\{{\frac{\partial i_{z}}{\partial z} + \frac{\partial i_{x}}{\partial z}} = {- {{v_{y}( \frac{1}{j\quad \omega \quad L\quad \Delta \quad d} )}.}}} & (2)\end{matrix}$

[0084] Combining equations (1) and (2) yields: $\begin{matrix}{{{\frac{\partial^{2}v_{y}}{\partial x^{2}} + \frac{\partial^{2}v_{y}}{\partial z^{2}} + {\beta^{2}v_{y}}} = 0},\quad {\beta = {- \frac{1}{\omega \sqrt{LC}\Delta \quad d}}}} & (3)\end{matrix}$

[0085] where β is the propagation constant. The phase and groupvelocities are antiparallel and are given by: $\begin{matrix}{{v_{\varphi} = {\frac{\omega}{\beta} = {{- \omega^{2}}\sqrt{{LC}\quad}\Delta \quad d}}},{v_{g} = {( \frac{\partial\beta}{\partial\omega} )^{- 1} = {{+ \omega^{2}}\sqrt{LC}\Delta \quad {d.}}}}} & (4)\end{matrix}$

[0086] Subsequently, a negative refractive index can be defined as:$\begin{matrix}{n = {\frac{c}{v_{\varphi}} = {- {\frac{1}{\omega^{2}\sqrt{LC}\sqrt{\mu_{0}ɛ_{0}}\Delta \quad d}.}}}} & (5)\end{matrix}$

[0087] It is interesting to note that it is possible to achieve the sameresult for the phase velocity and index of refraction if an equivalentnegative permittivity and permeability are defined as: $\begin{matrix}{{\mu_{eq} = {- \frac{1}{\omega^{2}C\quad \Delta \quad d}}},{ɛ_{eq} = {- \frac{1}{\omega^{2}L\quad \Delta \quad d}}},} & (6)\end{matrix}$

[0088] so that the total stored time-averaged energy, expressed by:$\begin{matrix}{{W = {{\frac{\partial( {\mu_{eq}\omega} )}{\partial\omega}{E}^{2}} + {\frac{\partial( {ɛ_{eq}\omega} )}{\partial\omega}{H}^{2}}}},} & (7)\end{matrix}$

[0089] remains positive.

[0090] In order to model the 2-D wave equation represented by equation(3), metamaterial including an array of unit cells, each as depicted inFIG. 2, was implemented in a microwave-circuit simulator simulator. Tosimulate the incidence of waves on this LHM in a circuit environment, aright-handed medium (RHM) was also designed. The topology of the unitcells in the RHM were identical to those of FIG. 2, except that theroles of C and L were interchanged, giving rise to the well-known LClow-pass network representation of free-space propagation.

[0091] The specification of the unit cell parameters in both mediaprovides information about the permissible operating frequencies, therelative refractive indices, and also the required inductance andcapacitance values. In both media, the wave impedance is given by:$\begin{matrix}{Z_{M} = {\sqrt{\frac{L}{C}}.}} & (8)\end{matrix}$

[0092] It is therefore reasonable to begin with the simple constraintthat the two media be matched, and moreover, matched to free space(Z_(M)=377 Ω).

[0093] In the limit Δd<<λ in equation (5), it is not possible to specifydirectly indices of refraction in the individual media, but it ispossible to specify a relative index of refraction through the ratio oftheir respective phase shifts per unit cell β_(LHM)Δd/β_(RHM)Δd. The LHMwas designed to be denser than the RHM, arbitrarily by a factor of 2.Specifically, phase shifts per unit cell of |β_(RHM)Δd|=0.25 and|β_(LHM)Δd|=0.5 were chosen in the right-handed and left-handed media,respectively. Choosing an operating frequency of 2 GHz, the LHM and RHMunit cell capacitive and inductive elements were specified as(C_(LHM)=422.5 fF, L_(LHM)=60.0 nH) and (C_(RHM)=52.8 fF, L_(RHM)=7.5nH), respectively. The corresponding cut-off frequencies are given by:$\begin{matrix}{{f_{c,{LHM}} = \frac{1}{4\pi \sqrt{L_{LHM}C_{LHM}}}},{f_{c,{RHM}} = \frac{1}{\pi \sqrt{L_{RHM}C_{RHM}}}},} & (9)\end{matrix}$

[0094] which were found to be 500 MHz and 16 GHz, respectively. Finally,the designed LHM and RHM arrays were appropriately terminated withmatching resistors on all edges and simulated with a microwave-circuitsimulator.

[0095] To verify the ray picture presented in the phase-matchingargument of FIG. 1, a RHM/LHM interface was constructed using 42×42 RHMand LHM unit cell arrays with β_(RHM)Δd=+0.25 and β_(LHM)Δd=−0.5,yielding a relative refractive index of −2. The RHM unit cell array wasexcited with a plane wave, which was simulated using sequentiallyphase-shifted voltage sources along the left boundary of the unit cellarray. FIG. 3 illustrates the plane wave illuminating the RHM/LHMinterface at 2 GH_(z) for an incident angle of θ_(RHM)=29° with the axesbeing labelled according to unit cell number and the right verticalscale designating radians. The steepest phase descent in the LHM isobserved along the direction of propagation, which is θ_(LHM)=14° fromthe normal, in exact correspondence with Snell's Law for the givendesign parameters.

[0096] Turning now to FIG. 4, it can be seen that a RHM/LHM planarinterface 410 can also be used to focus electromagnetic waves from apoint source located inside the RHM, within the LHM. These conditionscan be modelled by exciting a single node 400 inside the RHM andobserving the magnitude and phase of the voltages to ground at allpoints in the LHM. The focusing effect manifests itself as a “spot”distribution of voltage 420 at a predictable location in the LHM.

[0097] An alternative embodiment of an RHM/LHM interface used togenerate the results of FIG. 3 will now be examined. In this embodimentfinite length transmission line sections (k, Z₀, finite Δd) are insertedin each LC unit cell. In order that the host transmission-line mediumdoes not significantly alter the propagation constant predicted byequation (5), it is necessary to modify the loading elements C_(LHM) andL_(LHM) to compensate for the presence of the distributed transmissionline parameters. In the final design, the lines in each unit cell aredesigned to be air-filled, with Δd=5 mm. From equation (5), and usingthe appropriately compensated loading element values, the correspondingequivalent, absolute index of refraction of the LHM is approximatelyequal to −2.4. To maintain the relative refractive index of −2, theabsolute index of refraction of the RHM is made to be +1.2.

[0098] In the interest of completeness, cases of both positive andnegative refraction are examined, with the host transmission line mediumin place. In the first case, a 42×21 RHM unit cell array is interfacedwith another 42×63 RHM unit cell array with a relative refractive indexof +2. The plane wave source is placed 11 unit cells into the first RHM.Here, focusing is not expected since Snell's Law for positive-indexmedia predicts a continued divergence into the second RHM. FIG. 5 showsa point source illuminating the RHM/LHM interface at 2 GH_(z) as well asthe corresponding magnitude and phase plots of these results with theaxes labelled according to cell number. As can be seen, the resultsconfirm that the cylindrical wave excitation diverges into the secondmedium.

[0099] In the second case, a RHM unit cell array is interfaced with aLHM unit cell array with a relative refractive index of −2. The unitcell array dimensions and source location are as specified above. Thisarrangement is expected to show focusing inside the LHM, in accordancewith FIG. 4. The paraxial limit dictates a focus in the LHM at twice thedistance of the source from the interface, or near unit cell 44 of thearray. FIG. 6 shows the point source illuminating the RHM/LHM interfaceat 2 GH_(z), as well as the corresponding magnitude and phase resultswith the axes labelled according to cell number. As can be seen, theresults show focusing in the LHM, manifested in increased voltageamplitudes (nearly 65% of the source amplitude), and also in thereversal of the concavity of the wavefronts at both the RHM/LHM boundaryand the expected focal point.

[0100] For applications in which waves are propagated only in a singledimension e.g. phase-shifting lines as will be described, themetamaterial above may be simplified to a 1-D array using a singletransmission line i.e. a linear LHM metamaterial is a simplified case ofthe planar metamaterial.

[0101] Since the metamaterial exhibits left-handedness and allowselectromagnetic radiation to be guided and controlled, the metamaterialis useful in a number of applications. Examples of applications for thepresent metamaterial will now be described.

[0102] Near-Field Focusing Device

[0103] Turning now to FIG. 7, a focusing device is shown that includes a55 mm×50 mm parallel-plate waveguide serving as a RHM interfaced with anegative refractive index metamaterial including an 11×6 unit cell arrayaround 1.5 GHz. Each unit cell has a dimension of 5 mm. The fabricated11×6 unit cell areas includes 400 μm wide microstrip lines on a 60 milRexolite® dielectric (ε_(r)=2.53) substrate. Shunt chip inductors areembedded into 1 mm holes drilled into the substrate at the appropriateunit cell sites. Chip capacitors are surface-mounted between gaps etchedinto the grid lines, and additional capacitors are placed at the arrayedges to maintain unit cell uniformity throughout. The two media meetalong a flat border, defining their respective extents. The LHM extentis parallel to and displaced from the nearest microstrip line. Theparallel-plate waveguide is excited with a shorted vertical probeconnected to a SMA connector. Matching chip resistors of 100Ω are usedto terminate the unit cells at the LHM and parallel-plate waveguideboundaries. The focusing device as shown measures approximately 60 mm×95mm×1.5 mm.

[0104]FIG. 8 shows a sample of measured vertical E-field distributionsover the 11×6 LHM unit cell array at 1.55 GHz, 1.65 GHz, and 2.55 GHz.The first two frequencies lie in the LH passband, whereas the lastfrequency occurs in a second (RH) passband. In accordance with theory,the field distributions at 1.55 GHz and 1.65 GHz exhibit focusing,manifested by a localized region of increased transmission through thefocusing device. The maximum focal amplitude obtained was observed at afrequency of 1.65 GHz (the central plot of FIG. 8), where thedistinction between the peak and the two edges of the spot was noted tobe approximately 15 dB. Furthermore, the concavities of the wavefrontsof FIG. 8 also indicate the convergent progression of phase associatedwith focusing. As expected, neither phenomenon is evident at 2.55 GHz,since the second passband does not possess left-handed properties. Aswill be appreciated, the results demonstrate focusing of an incidentcylindrical wave over an electrically short area, i.e. near-fieldfocusing.

[0105] Planar Resonance Cone Metamaterial

[0106] Turning now to FIGS. 9 to 11, a planar resonance conemetamaterial is illustrated. In this embodiment, the metamaterialincludes a substantially orthogonal, coplanar, periodic square-celledtransmission line network that is analogous to a sheet of uniaxialplasma with permittivities of opposite signs in two orthogonal, in-planedirections. As can be seen, the transmission line network is disposedover a ground plate and parallel to it. Electromagnetic radiation is fedto the transmission line network at one corner with respect to theground plate. The transmission line network is connected to instrumentsthrough coaxial connectors as shown in FIG. 10 and includes chipcapacitors and meander-line inductors as shown in FIG. 11. Resonancecone angle scans with frequency and with peak frequencies detected atthe mid-points of the transmission line network sides, yielded anend-to-end frequency ratio of 2:1 with a transmission line network ofonly 4 cells by 4 cells. A center frequency of 1.4 GHz was used. Thecommercial chip capacitive and inductive elements used to construct thetransmission line network yielded a compact array. A larger transmissionline network having 24 cells by 24 cells was also modelledcomputationally. This transmission line network displayed resonancecones that showed little or no near-field beam spreading with increasingdistance from the source. When a cone beam encountered an edge of thetransmission line network with insufficient resistive absorption,specular reflection of the cone beam was observed.

[0107] Changing the reactive loads creates a different transmission linenetwork. For example, a “transpose” transmission line network can becreated by interchanging the inductive and capacitive elements.

[0108] Phase-Shifting Line Using a Linear Metamaterial

[0109] The metamaterial in accordance with the present invention is alsosuitable for use in a compact 1-D phase-shifting device that can be usedto synthesize arbitrary transmission phases. Such as shifting deviceincludes sections of LHM metamaterial lines cascaded with sections ofconventional RHM transmission lines. Several embodiments of such aphase-shifting device, having a 1, 2, 4, and 8 stages can be seen inFIG. 12. The phase-shifting device offers significant advantages whencompared to standard delay transmission lines. The phase-shifting deviceis more compact in size and can achieve a positive, a zero, or anegative phase shift while occupying the same or shorter physicallength. The phase-shifting device also exhibits a linear, flatter phaseresponse with frequency, leading to shorter group delays.

[0110] In a LHM, the phase leads in the direction of group velocity,therefore incurring a positive phase shift with propagation away fromthe source. In addition, it is well known that in conventional RHMtransmission lines, the phase lags in the direction of positive groupvelocity, thus incurring a negative phase shift with propagation awayfrom the source. It therefore follows that phase compensation can beachieved at a given frequency by cascading a section of a LHM with asection of a RHM to form a unit cell with a given phase shift.

[0111] The unit cell of the phase-shifting device is shown in FIG. 13and has total dimension d_(o). The unit cell includes a hosttransmission line medium TL periodically loaded with discrete lumpedelement components, L_(o) and C_(o).

[0112] The dispersion characteristics of the unit cell can be determinedby the following equations: $\begin{matrix}{{\cos ( {\beta_{Bloch}d} )} = {{{\cos ( {\beta_{TL}d} )}( {1 - \frac{1}{4\omega^{2}L_{o}C_{o}}} )} + {{\sin ( {\beta_{TL}d} )}( {\frac{1}{\omega \quad C_{o}Z_{o}} + \frac{Z_{o}}{\omega \quad L_{o}}} )} - \frac{1}{4\omega^{2}L_{o}C_{o}}}} & (10)\end{matrix}$

[0113] where:

[0114] β_(Bloch) is the Bloch propagation constant; and

[0115] β_(TL)=ω{square root}{square root over (LC)} is the propagationconstant of the host transmission line.

[0116] In order to consider a series of cascaded unit cells as aneffective periodic medium, the physical length of the unit cell must bemuch smaller than a wavelength, therefore restricting the phase shiftper unit cell and the length of the transmission lines to small values(βBlochd<<1 and βTLd<<1). Thus, an effective propagation constant, βeff,can be defined for the medium by the following equation: $\begin{matrix}{\beta_{eff} = {\pm \sqrt{( {L - \frac{1}{\omega^{2}C_{o}d}} )( {C - \frac{1}{\omega^{2}L_{o}d}} )}}} & (11)\end{matrix}$

[0117]FIG. 14 shows the dispersion relation for a unit cell with typicalparameters C_(o)=15 pF, L_(o)=20 nH, Z_(o)=50Ω and β_(TL)d=0.28 rad. Itcan be observed that the corresponding dispersion diagram exhibits aband structure with two distinct passbands and stopbands. Expressionsfor the pertinent cut-off frequencies as indicated in FIG. 14 are asfollows: $\begin{matrix}{{f_{b} = \frac{1}{4\pi \sqrt{L_{o}C_{o}}}},{f_{c1} = \frac{1}{2\pi \sqrt{L\quad C_{o}}}},{f_{c2} = \frac{1}{2\pi \sqrt{L_{o}C}}}} & (12)\end{matrix}$

[0118] Equating f_(c1) and f_(c2) causes the stopband between these twocut-off frequencies to close and the band becomes continuous. Thiscorresponds to the following matching condition for the LHM and RHMsections:

Z _(o) ={square root}{square root over (L_(o)/C_(o))}={squareroot}{square root over (L/C)}  (13)

[0119] Under this condition, it can be shown that the effectivepropagation constant of equation (11) simplifies to the expression:$\begin{matrix}{\beta_{eff} \approx {{\omega \sqrt{LC}} + \frac{- 1}{\omega \sqrt{L_{o}C_{o}}}}} & (14)\end{matrix}$

[0120] Expression (14) can be interpreted as the sum of the propagationconstants of the host transmission line and a uniform LHM line. For atransmission line medium with periodicity d and phase shiftφ_(TL)=β_(TL)d, the total phase shift per unit cell, |φ_(o)|, given thatthe matching condition of equation (13) is satisfied, can be written as:$\begin{matrix}{{\varphi_{o}} = {{\beta_{eff}d} = {\varphi_{TL} + \frac{- 1}{\omega \sqrt{L_{o}C_{o}}}}}} & (15)\end{matrix}$

[0121] Thus equations (13) and (15) can be used to determine uniquevalues for L_(o) and C_(o) for any phase shift, φ_(o), given atransmission line section with intrinsic phase shift β_(TL) andcharacteristic impedance Z_(o).

[0122] The unit cell of FIG. 13 was implemented in coplanar-waveguide(CPW) technology at a design frequency of f_(o)=0.9 GHz usingmicrowave-circuit simulator. The lumped element components and thetransmission lines were assumed to be ideal. The transmission lines weredesigned with Z_(o)=50Ω, ε_(r)=2.2, dielectric height, h=20 mils andlength d=7.4 mm. The dispersion relation for the unit cell with φ_(o)=0°is shown in FIG. 15. The impedance matching condition of equation (13)has been satisfied and therefore, the stopband has been closed. It canalso be observed that at f_(o)=0.9 GHz, the phase shift is 0°.

[0123] Based on the above parameters a +10-degree phase-shifting devicewas designed including 4 unit cells of total physical length 0.11λCPW=32mm, with loading element values of C_(o)=21.66 pF and L_(o)=54.15 nH.The resulting phase characteristic of the phase-shifting device is shownin FIG. 16. It can be observed that the phase shift is +10 degrees atf_(o)=0.9 GHz with a phase slope of −95.7-degrees/GHz. In addition, thephase response around fo is linear with respect to frequency, implying aconstant group delay for the phase-shifting device. Also shown in FIG.16 is the phase characteristic of a conventional 50Ω transmission lineof length 35/36 λCPW=275.7 mm for comparison. The transmission lineexhibits as expected a −350-degree phase shift at fo, which isequivalent to +10 degrees, with a phase slope of −389-degrees/GHz. Inaddition, its physical length is approximately 9 times longer than the 4unit cell phase-shifting line.

[0124] Referring back to FIG. 12, an impression of the relative sizes ofthe two structures can be obtained. The proposed +10-degreephase-shifting device offers two main advantages over its equivalenttransmission line implementation, namely a +10-degree phase shift whileoccupying significantly less physical space than a conventionaltransmission line, and a shorter group delay which is desirable forapplications that require broadband operation.

[0125] Also shown in FIG. 16 is the phase characteristic of a 4-stage−10° phase-shifting device with loading element values of C_(o)=29.46 pFand L_(o)=73.65 nH. It can be observed that the −10° phasecharacteristic is similar to the +10° phase characteristic, with a phaseslope of −72.9°/GHz. The −10° phase-shifting device occupies the samephysical length as the +10° phase-shifting device and has loadingelement values that are at a ratio of approximately 1.4:1 compared tothe ones used for the +10° phase shifting device. This implies that areasonable variation of the loading element values using tuneablecomponents can produce either a positive or a negative phase shift.

[0126]FIG. 16 also shows the phase characteristic of a 4-stage +10°uniform backward wave L-C line with loading element values ofC_(o)=81.06 pF and L_(o)=202.64 nH. This corresponds to the unit cell ofFIG. 13 with the transmission line sections removed. Setting thereforeφTL to zero in equation (15), implies that φ_(o) will always remainnegative, corresponding to a phase advance for a positively travelingplane wave of the form exp(−jβeffd). The absence of transmission linesimplies that any combination of C_(o) and L_(o) values will produce aphase shift that will always be greater than 0 degrees for the unitcell. This is verified by the phase characteristic of the L-C line inFIG. 16 that stays well above 0 degrees at all times. Therefore,although the backward wave L-C line and the proposed unit cell of FIG.13 can both provide positive phase shifts, the latter has the additionaladvantage of being able to provide a negative phase shift by simplyvarying the loading element values. More importantly, short, broadband0-degree phase shift lines can be realized which can be used in numerousapplications.

[0127] The 1-D phase-shifting structures were constructed using CPWtechnology on a Rogers™ RT5880 substrate with εr=2.2, and a dielectricheight, h=20 mils. Standard size 0402 Panasonic ECJ-0EC capacitors wereused for C_(o) with dimensions L=1 mm, W=0.5 mm, and H=0.5 mm, andstandard size 0603 Panasonic ELJ-RE/FJ inductors were used for L_(o)with dimensions L=1.6 mm, W=0.8 mm, and H=0.8 mm. The 50Ω transmissionlines were designed to have a length of d=7.4 mm, and a gap of s=0.6 mmwas created to accommodate the series capacitors, resulting in a unitcell of total length d_(o)=8 mm. At 0.9 GHz the transmission lines weredesigned with a center conductor width of w=4 mm and a gap between thecenter conductor and the ground plane of g=0.106 mm. In addition, two 5mm transmission lines were added to each end of the phase-shiftingdevices in order to accommodate SMA connectors. A Thru-Reflect-Line(TRL) calibration was carried out in order to remove the effects of theconnectors and establish the reference planes at the input and outputports of the phase-shifting devices. All measurements were carried outon a HP8753C series vector network analyzer. The simulated results wereobtained by replacing the ideal models for the capacitors and inductorswith their corresponding S-parameter files provided by the vendors.Since the S-parameter files were extracted directly from devicemeasurements, they therefore contain all the parasitic values associatedwith each component.

[0128]FIG. 17 shows the simulated ( - - - ) and measured (-) phaseresponses of 1-stage, 4-stage and 8-stage 0° phase-shifting devices withloading element values of C_(o)=12 pF and L_(o)=100 nH. It can beobserved that the experimental results match very closely the simulatedresults, with phase slopes of −30.2°/GHz, −122.1°/GHz and −237.5°/GHzfor the phase-shifting devices respectively. The measured insertionlosses are 0.1 dB, 0.7 dB and 1.6 dB respectively at f_(o)=0.9 GHz. Itis therefore evident that any number of stages can be used in order toproduce the desired phase shift, however as the number of stagesincreases, so does the slope of the phase characteristic. For broadbandapplications therefore, it is desirable to keep the number of stages toa minimum.

[0129]FIG. 18 shows the simulated ( - - - ) and measured (-) phaseresponses of a 4-stage +10° phase-shifting device with loading elementvalues of C_(o)=12 pF and L_(o)=68 nH and a 4-stage −10° phase-shiftingdevice with loading element values of C_(o)=15 pF and L_(o)=120 nH. Itcan be observed that the experimental results correspond very closely tothe simulated results with phase slopes of −136.6°/GHz and −108.4°/GHzfor the +10° and −10° phase-shifting devices respectively. Also shown inFIG. 18 is the magnitude response of the −10° 4-stage phase-shiftingdevice. The simulated ( - - - ) and measured (-) results correspondclosely, however as the number of stages is increased the phase-shiftingdevice begins to radiate, and resembles a leaky-wave structure.Therefore the phase-shifting devices should be kept short in length toavoid unnecessary energy leakage. The measured insertion losses atf_(o)=0.9 GHz are 0.4 dB and 0.5 dB respectively for the +10° and −10°phase-shifting devices.

[0130] The phase-shifting devices described above use cascaded sectionsof LHM lines and conventional transmission lines and offer significantadvantages over conventional delay lines and uniform backward-wave L-Clines. The phase-shifting devices are compact in size, can be easilyfabricated using standard etching techniques and exhibit a linear phaseresponse around the design frequency. The phase-shifting devices canincur a negative or a positive phase, as well as a 0° phase depending onthe values of the loading elements, while maintaining a short physicallength. In addition, the phase incurred is independent of the length ofthe phase-shifting devices. Due to the compact, planar design, thephase-shifting devices are easily integrated with other microwavecomponents and devices making them ideal for broadband applicationsrequiring small, versatile, linear phase shifters.

[0131] Spectrum Analyser Using a Resonance Cone Metamaterial

[0132] The resonance cone metamaterial is useful in spectrum analysis.To illustrate this, a spectrum analyser incorporating resonance conemetamaterial was simulated. For the representation of metamaterial,analyser simulations employed a full-electromagnetic, thin-wiremoment-method program, permitting the insertion of lumped circuitelements in finite-length wire segments. The basic network simulated isshown in FIG. 21, and includes a 12×12 unit cell array, each unit cellbeing a 2.5 mm square, yielding a total grid size of 30 mm×30 mm. Thegrid is disposed above the ground plane a distance of 2.5 mm. Thecapacitance was 2 pF per segment and the inductance was 5.6 nH persegment. The chosen capacitance and inductance values were selected foravailability as well as to achieve the desired resonance cone effectsusing existing X-Y scanning and network analysis equipment. Theinductors used in the experiment had a manufacturer nominal qualityfactor Q of around 27. This was achieved in the simulation by adding a1.6 Ω resistor in series with each inductor. A resistor of the samevalue was added in series with each capacitor as well. Also shown inFIG. 21 is a 100 kΩ resistor from each grid intersection to ground toenable deduction of the grid-to-ground voltage from the computedresistor current. Along the edges, 50 Ω resistors were connected toground in place of the 100 kΩ resistors. For this value of edgeresistance, resonance cone specular reflection from the edges is notsignificant.

[0133] Contour plots of grid-to-ground voltage magnitude and thePoynting vector real parts evaluated from the electric and magneticfields on the ground-plane level at the center of each unit cell for acorner feed of 1 volt are shown in FIG. 22. The simulated resonancecones and the way the cone orientation scans with frequency over theeight frequencies employed are illustrated. The arrows depict thePoynting vector real parts calculated on the ground plane. From thePoynting vector plot it can be seen that power flow is directed alongthe resonance cones. Apparent specular reflection from the grid edges issometimes visible (for example at 0.7, 0.9, 1.9 and 2.3 GHZ) butgenerally it is weak. As well, very weak parasitic resonance cones canbe seen in FIG. 22 (for example at 0.9 and 2.3 GHZ), probably caused byscattering from the inhomogeneities inherent in the grid structure. FIG.23 shows the frequency response for the voltages across the 50 Ωresistors to ground at all points along two edges of the grid.

[0134] Focusing Device Using Two Resonance Cone Metamaterials

[0135] Another focusing device making use of resonance cone metamaterialwas created using a two media plane including metamaterial similar tothat shown in FIG. 11. By interchanging (transposing) the inductors andcapacitors in the grid in part of the media, an interface is formed,separating two media. If the plasma negative and positive permittivitiesare interpreted by representing the media in terms of arrays ofinductors and capacitors respectively, then interchanging the inductorsand capacitors corresponds to moving the operating point from onehyperbolic region to the other in the parameter space as shown in FIG.20. The hyperbolic regions characterize the relevant partialdifferential equation. The vertical axis involves the electron cyclotronfrequency ω_(c) and is proportional to the square of the ambientmagnetic field, while the horizontal axis involves the plasma frequencyω_(p) and is proportional to the ambient electron density. The capacitorand inductor symbols characterize the impedance properties of the plasmamedium as they effect the input of impedance of a small, spherical RFprobe immersed in the plasma medium. These two hyperbolic regionssuggest two quite different periodic physical media that, wheninterfaced, display unusual refraction phenomena. If the inductors andcapacitors are regarded as physical entities and not just as aids tointerpretation, a physical inductor-capacitor (L-C) network becomes ametamaterial that exhibits a resonance cone phenomena. This enables atwo-medium configuration with a linear interface between the originalmedium and the transpose medium.

[0136] The component layout of a focusing device includingmulti-resonance cone materials is shown in FIG. 24, and its physicalgrid realization is shown in FIG. 25 in which the grid elements arechip-type surface-mount inductors and capacitors soldered together. Thegrid has 12 by 12 unit cells. Medium 1 is constituted by the lower 5rows of the grid and includes inductors on the horizontal segments andcapacitors on the vertical segments. Medium 2 is constituted by theupper 6 rows of the grid and includes capacitors on the horizontalsegments and inductors on the vertical segments. The sixth row of cellsis the transition region, or interface, where the lower edges of theunit cell carry inductive loads, the upper edges carry capacitive loads,and the vertical segments carry a capacitor on the lower half and aninductor on the upper half. Therefore the boundary between the two medialies in the middle of the transition region.

[0137] To accommodate the physical size of the chip-type inductors andcapacitors, the unit cells in the transition region are elongated i.e.the vertical dimension is doubled compared to unit cells elsewhere inthe grid. The overall size of the grid is 30 mm×32.5 mm and the boundaryis a horizontal line 15 mm from the bottom of the grid. The grid isaligned so that individual vertical transmission lines run directlythrough both media and the interface. In the refraction experiments, thevertical E-field is picked up by an open-ended coaxial probe positionedsequentially just above each conductor intersection, and the probesignal is fed to a network analyzer with output S21 which, when plottedon a linear scale, is approximately proportional to the grid-to-groundvoltage.

[0138] The two-medium result for corner feed with the feed point beingbelow point A in FIG. 24, is shown in FIG. 26, which displays clearlythe negative refraction of the resonance cone as it traverses thetransposition interface. Experimental results at the top exhibitcontours of normalized /S21/, and the simulation at the bottom exhibitscontours of grid-to-ground voltage magnitude with the arrows depictingthe Poynting vector real parts on the ground plane. The counter lines inthe bottom plot are at the same levels as the three lowest contour linesin the top plot. Notice that specular reflection from the interface isnegligible and there is no visible transmission into the second mediumin the same direction as the incident cone. This can also be seen fromthe Poynting vectors in the bottom plot of FIG. 26, which displays thesimulation results, where the power flow associated with specularreflection from the interface is negligible and no power along theoriginal resonance cone is transmitted into the second medium. In orderto extend the refraction results to include focusing, the physicalset-up of FIG. 25 was used with coaxial feed from beneath the groundplane to point B on the grid in FIG. 24, near the middle of one of thetwo planar metamaterials.

[0139] Simulation and experimental results are shown in FIGS. 27 and 28at 1.3 GH_(z). The arrows display the Poynting vector real portscalculated on the ground plane. Heavy contour is at 0.707 of focalmaximum of 0.867 units of normalized /S21/. Simulation and experimentalresults are also shown in FIGS. 29 and 30 at 1.7 GH_(z). In FIG. 29heavy contour is at 0.707 of focal maximum of 0.708 and in FIG. 30 heavycontour is at 0.707 of focal maximum of 0.764. In all cases, one can seethe cones emanating from the feed point (the “source”), the backwardrefraction at the y=15 mm interface, and the cones merging at the“focus”. Note that in FIGS. 27 and 29 the power flow closely follows thedirections of the resonance cones. Furthermore, FIG. 29 shows the powerflow emanating from the source and following the resonance cones toconverge at the focus, from where it diverges again into a new set ofresonance cones. In FIGS. 29 and 30, the focal region is taken to bebounded by the contour that is at 0.707 of the focal region maximum, sothis contour may be termed the “half-power” contour. It is worth noting,that at 1.7 GHz, the experimental focal region boundary lies within asquare measuring {fraction (1/25)}th free-space wavelength on a side, sothe phenomenon properly may be termed “sub-wavelength focusing”. Incontrast, at 1.3 GHz the focal point is too close to the upper edge ofthe material to exhibit an equally well localized focal region, and thusthis represents a useful low-frequency limiting case for the particularoverall grid size in use. As well, with increasing frequency, it shouldbe noted that the focal region moves gradually toward the boundarybetween the two media.

[0140] Couplers Using Linear Metamaterials

[0141] The present metamaterial is also useful in couplers by combiningthe metamaterial with conventional microchip (MS) transmission lines. Inparticular, it has been found that the metamaterial is useful in both acoupled-line coupler and a branch-line coupler. The MS/LHM coupled-linecoupler is realized by means of one regular microstrip line that isedge-coupled to a LHM line. Such a coupler excites modes on the twolines whose propagation constants are co-directional but whosecorresponding Poynting vectors are contra-directional, leading tobackward power coupling. Moreover, by comparison to a conventionalcoupled-line microstrip coupler of the same length and line spacing, thepresent MS/LHM coupled-line coupler offers better performance in termsof coupled power, return loss and isolation without any bandwidthdegradation. The MS/LHM coupled-line coupler can also be designed forarbitrary backward coupling.

[0142] The branch-line coupler uses LHM lines to split power equallybetween the output ports with interesting phase compensationcharacteristics. In conventional branch-line couplers, the through andcoupled ports are −90° and −180° out of phase respectively withreference to the input port. The present branch-line coupler using anLHM line, results in the power emerging from the coupled port to be inphase with the input port and permits a choice of either positive ornegative phase-quadrature (±90°) at the through port.

[0143] For ease of understanding, the theory behind the coupled-linecoupler will be described. A MS/LHM coupled-line coupler can be realizedby means of one regular microstrip line that is edge-coupled to a LHMline as shown in FIG. 31. The LHM is synthesized by periodically loadinga host microstrip transmission-line (TL) medium with series capacitorsand shunt inductors. In such a backward-wave transmission line, thephase velocity and power flow are anti-parallel.

[0144] Coupled-line couplers rely on proximity interaction of fieldsbetween transmission lines to transfer power between them. Inco-directional coupled-line couplers, the direction of the propagationvectors on the two lines is the same. When such couplers are constructedusing either two regular or two LHM lines, the underlying propagationphenomenon is fundamentally similar to that on a corresponding isolatedline of the same type. Indeed in such co-directional couplers includingtwo lines of the same type, the power also couples co-directionally andappears at the far end of the coupler (i.e. away from the source).Peculiar features arise when two different types of lines are used.Herein it is shown that a MS/LHM coupler excites co-directional forwardtraveling waves but delivers power backwards as a result of the powerflow being contra-directional.

[0145] The analysis presented herein assumes that port 1 in FIG. 31(a)is the excited port. The operation of coupled line couplers can beunderstood in terms of constructive and destructive interference of thetwo modes that are allowed to propagate. Using a coupled-modedifferential-equation formulation, it is possible to show that thegeneral expressions for voltages and currents on the two lines take theform:

V ₁=(V _(c) ⁺ e ^(−jβ) ^(_(c)) ^(z) +V _(c) ⁻ e ^(jβ) ^(_(c)) ^(z))+(V_(π) ⁺ e ^(−jβ) ^(_(π)) ^(z) +V _(π) ⁻ e ^(jβ) ^(_(π)) ^(z))  (16)

V ₂ =R _(c)(V _(c) ⁺ e ^(−jβ) ^(_(c)) ^(z) +V _(c) ⁻ e ^(jβ) ^(_(c))^(z))+R _(π)(V _(π) ⁺ e ^(−jβ) ^(_(π)) ^(z) +V _(π) ⁻ e ^(jβ) ^(_(π))^(z))  (17)

[0146] $\begin{matrix}{I_{1} = {{\frac{1}{Z_{c1}}( {{V_{c}^{+}^{{- {j\beta}_{c}}z}} - {V_{c}^{-}^{{j\beta}_{c}z}}} )} + {\frac{1}{Z_{\pi 1}}( {{V_{\pi}^{+}^{{- {j\beta}_{\pi}}z}} - {V_{\pi}^{-}^{j\quad \beta_{\pi}z}}} )}}} & (18) \\{I_{2} = {{\frac{R_{c}}{Z_{c2}}( {{V_{c}^{+}^{{- {j\beta}_{c}}z}} - {V_{c}^{-}^{{j\beta}_{c}z}}} )} + {\frac{R_{\pi}}{Z_{\pi 2}}( {{V_{\pi}^{+}^{{- {j\beta}_{\pi}}z}} - {V_{\pi}^{-}^{{j\beta}_{\pi}z}}} )}}} & (19)\end{matrix}$

[0147] Subscripts 1 and 2 refer to line 1 (microstrip) and line 2 (LHM)of the coupler respectively; c and π are the two possible modes and thecoefficients R_(c) and R_(π) are the ratios of the voltages on line 2 toline 1 for each mode.

[0148] To analyze the MS/LHM coupler of FIG. 31, some simplifyingassumptions are made. First, the lines of the coupler are assumed to thefar removed from each other so that the coupled-mode propagationconstants β_(c) and β_(π) correspond to the isolated propagationconstants β₁ and β₂ respectively of each line. This implies that β₂ (LHMline) has a negative value. Hence all occurrences of β_(c) and β_(π) inequations (16) to (19) can be replaced by β₁ and β₂ respectively.Furthermore line 1 is assumed to be the excited line and therefore, itis anticipated that mode 1 will be the dominant mode for determining thevoltage on line 1. Second, to minimize reflections, only the case ofproperly terminated lines is considered. Thus each mode travels in onedirection only. Mode 1, being dominant on line 1, has a forward phasevelocity and carries power away from the source. The dilemma lies in thechoice of the direction of the phase velocity for mode 2. To resolvethis issue, one can consider the limiting case when the magnitudes ofthe isolated propagation constants of the modes are close to each other.As mode 1 travels forward, due to phase matching constraints, it isexpected that the coupled mode (mode 2 in this case) should also travelforward. The present situation is the reciprocal case and therefore,power from the excited regular MS line will couple backwards into port 2of the LHM line (see FIG. 31(a)). In the general case when the two modepropagation constants are not close to each other, maximum power can becoupled backwards into port 2 (see FIG. 32(a)) only when the excitedmodes are co-directional. Hence in the following discussion, only thecase of co-directional (phase-wise) coupled modes will be considered.

[0149] Lastly, it is assumed that the magnitudes of the impedances foreach mode on line 1 are the same and are denoted by Z₁; similarly forline 2 they are denoted by Z₂. Caution must be exercised in choosing thesigns of the impedances in the current equations 3 and 4. The impedanceterm in front of V_(c) ⁺ and V_(c) ⁻ (equation 3) will be positive asusual. Since V_(π) ⁺ and V_(π) ⁻ (equation 3) actually correspond tobackward and forward traveling waves respectively on line 1 (due to thenegative sign of β₂), and to ensure that the power carried by a forwardtraveling wave in a regular medium is also forward, Z_(π1) should benegative. Similar arguments applied to line 2 (LHM line) require thatthe sign of Z_(c2) be negative and that of Z_(π2) be positive to ensurethat power flow and phase flow occur in opposite directions.

[0150] With these assumptions, equations (16) to (19) can be simplifiedto:

V ₁ =V _(c) ⁺ e ^(−jβ) ^(₁) ^(z) +V _(π) ⁻ e ^(jβ) ^(₂) ^(z)  (20)

V ₂ =R _(c) V _(c) ⁺ e ^(−jβ) ^(₁) ^(z) +R _(π) V _(π) ⁻ e ^(jβ) ^(₂)^(z)  (21)

[0151] $\begin{matrix}{I_{1} = {{\frac{V_{c}^{+}}{Z_{1}}^{{- {j\beta}_{1}}z}} + {\frac{V_{\pi}^{-}}{Z_{1}}^{j\quad \beta_{2}z}}}} & (22) \\{I_{2} = {{{- \frac{R_{c}}{Z_{2}}}V_{c}^{+}^{{- {j\beta}_{1}}z}} - {\frac{R_{\pi}}{Z_{2}}V_{\pi}^{-}^{{j\beta}_{2}z}}}} & (23)\end{matrix}$

[0152] Applying boundary conditions at port 1 (presence of source V_(s)and matching source impedance Z₁) and port 4 (termination to Z₂)respectively yields: $\begin{matrix}{{{V_{c}^{+}^{{j\beta}_{1}d}} + {V_{\pi}^{-}^{{- {j\beta}_{2}}d}}} = \frac{V_{s}}{2}} & (24)\end{matrix}$

 R _(c) V _(c) ⁺ +R _(π) V _(π) ⁻=0  (25)

[0153] Solving for V_(c) ⁺ and V_(π) ⁻, and plugging back into equations(20) and (21), the following expressions for the line voltages along thecoupler can be obtained: $\begin{matrix}{{V_{1}(z)} = {\frac{V_{s}}{2}\frac{{R_{\pi}^{{- {j\beta}_{1}}z}} - {R_{c}^{{j\beta}_{2}z}}}{{R_{\pi}^{{j\beta}_{1}d}} - {R_{c}^{{- {j\beta}_{2}}d}}}}} & (26) \\{{V_{2}(z)} = {\frac{V_{s}}{2}\frac{R_{\pi}{R_{c}( {^{{- {j\beta}_{1}}z} - ^{{j\beta}_{2}z}} )}}{{R_{\pi}^{{j\beta}_{1}d}} - {R_{c}^{{- {j\beta}_{2}}d}}}}} & (27)\end{matrix}$

[0154] Equations (26) and (27) confirm that port 4 (z=0) is the isolatedport and port 2 (z=−d) is the coupled port. Expressions for powerdelivered to the through port (port 3) and coupled port (port 2) can bederived from equations (26) and (27): $\begin{matrix}{P_{2} = {\frac{1}{Z_{2}}( \frac{V_{s}}{2} )^{2}\frac{( {R_{C}R_{\pi}} )^{2}( {1 - {\cos ( {( {\beta_{1} + \beta_{2}} )d} )}} )}{( {R_{c}^{2} + R_{\pi}^{2} - {2R_{c}R_{\pi}\cos \quad ( {( {\beta_{1} + \beta_{2}} )d} )}} )}}} & (28) \\{P_{3} = {\frac{1}{2Z_{1}}( \frac{V_{s}}{2} )^{2}\frac{( {R_{C} - R_{\pi}} )^{2}}{( {R_{c}^{2} + R_{\pi}^{2} - {2R_{c}R_{\pi}\cos \quad ( {( {\beta_{1} + \beta_{2}} )d} )}} )}}} & (29)\end{matrix}$

[0155] From equation (29) it can be seen that the power delivered to thethrough port (port 3) is minimized (i.e. the power to the coupled portis maximized) when the argument of the cosine function in thedenominator takes values which are odd integer multiples of π. Thisallows the definition of the coherence length of the coupler, which isthe optimal length for maximum coupling, as:

d _(coherence)=π/|β₁+β₂|  (30)

[0156] Note that in the above equation (30), β₂ is negative. Aninteresting feature of such a coupler is that it is possible to achievearbitrary backward coupling by making the magnitudes of the propagationconstants of the two modes close to each other.

[0157] Two coupled-line couplers were constructed, namely a short MS/LHMcoupled-line coupler and a long MS/LHM coupled-line coupler witharbitrary coupling. The substrate used to construct the couplers was alow-loss Rogers™ TMM4® having a relative dielectric constant ε_(r)=4.6and 50 mils thickness (loss tangent=0.002). The regular MS lines weredesigned using a microwave-circuit simulator, whereas Matlab®simulations of the dispersion relation of the periodic LHM line wereperformed to estimate the corresponding propagation constant at thedesign frequency. Moreover, microwave circuit simulations of thecouplers were performed using a microwave-circuit simulator.

[0158] The short MS/LHM coupled-line coupler was designed with a finitecoherence length to demonstrate the validity of the theory. The longMS/LHM coupled-line coupler has magnitudes of the two modalpropagation-constants close to each other. By taking a finite number ofcells (the optimal length infinite as defined by equation 15), acomparison with a conventional (MS/MS) coupled-line microstrip couplerof the same length, line spacing, and propagation constant is permitted.

[0159] Using a microwave circuit simulator the widths of the lines ofthe short MS/LHM coupled-line coupler are designed so that both the MSand LHM lines have an impedance of 50Ω when placed 0.4 mm apart. Thewidth of the MS and LHM lines used was 2.383 mm. Using a unit cell sizeof 5 mm for the LHM line, the choice of the loading elements are 2.7 nHshunt inductors and 0.9 pF series capacitors. Again, the microwavecircuit simulator is used to determine the effective dielectric constantfor the line segments. This is used in Matlab simulations of theperiodically loaded transmission-line based LHM line to extract itsdispersion characteristics (see FIG. 32(a)). A design frequency of 2.2GHz is selected for this configuration so that the magnitude of theisolated propagation constants differ enough (see FIG. 32(a)) to give ashort coherence length (see equation 15).

[0160] A spatial Fourier transform of the line voltages, extracted byexciting port 1 in ADS simulations, is shown for both lines of thecoupled-line coupler at 2.2 GHz using 400 sample points (see FIGS. 33(a)and (b)). These plots demonstrate the validity of the assumption thatthe propagation constants of the coupled modes (|β₁|=85.3 m⁻¹ and|β₂|=195 m⁻¹) are indeed close to the isolated propagation constants ofthe individual lines (compare to FIG. 32(a)). Moreover the Fourierspectrum demonstrates that both modes propagate forward (co-directional)and that β₁ is dominant on line 1 (MS). Furthermore, the assumption ofminimal reflections is evident from the small peaks in the negativeregion of the spectrum in FIGS. 33(a) and 33(b). In addition, usingequation (30), the coherence length is calculated to be 28.6 mm and canbe approximated by 6 unit cells of the MS/LHM coupled-line coupler. Bycomparing the coupled power as a function of frequency for variouscoupler sizes, this is verified to be true as shown in FIG. 34.Moreover, the input power at port 1 is V_(s) ²/8Z₀ and the output powerat the coupled port is given by equation (28). The implicated voltageratios R_(c) and R_(π) can be estimated from FIGS. 33(a) and 33(b) andthe corresponding theoretical value for the coupled power (S₂₁) amountsto −7.46 dB at the coherence length, which is close to the simulationresults for a 6 unit cell long coupler (see FIG. 34).

[0161] Fair comparison between a conventional coupled-line backwardcoupler (MS/MS) and a metamaterial long MS/LHM coupled-line coupler (seeFIG. 32(b)) can be made if their propagation constants and lineimpedances are similar. This is because, it is always possible to slowdown the coupled waves and improve coupling in conventional couplers byloading them. For this purpose, a design frequency of 2.8 GHz (FIG.32(a)) is selected. The widths of the MS and LHM strips of the MS/LHMcoupled-line coupler are taken to be 2.45 mm and 2.0 mm respectively sothat at a line separation of 0.4 mm at the design frequency, theimpedances of each line are close to 50Ω. The loading elements are thesame as above with reference to the short MS/LHM coupled-line coupler.

[0162] A regular MS/MS backward coupler was also designed at the samefrequency with a propagation constant of 108.2 m⁻¹ and impedance closeto 50 Ω. The widths of the strips were taken to be 2.122 mm in this caseand the length of the coupler was optimized to 14.5 mm (λ/4).

[0163] From MATLAB® simulations of the periodic LHM lines (see FIG.32(a)) it is found that at 2.8 GHz, the propagation constants for bothlines of the MS/LHM coupled-line coupler are similar to those of theMS/MS coupler lines (MS and LHM propagation constants are 108.9 m⁻¹ and105 m⁻¹ respectively). This makes the coherence length for the MS/LHMcoupled-line coupler very long (equation 15). For comparison with thedesigned conventional MS/MS coupler, the former is constructed out of 3unit cells corresponding to a total length of 15 mm (see FIG. 32(b)).

[0164] From theory (equation (15)) it is anticipated that this MS/LHMcoupled-line coupler will have a long coherence length if thepropagation constants of its two modes are close to each other. This istrue at 2.8 GHz as shown in the dispersion diagram for the LHM-line (seeFIG. 31(a)).

[0165] For comparison with a conventional MS/MS backward coupler,simulation and experimental results are presented in FIGS. 35(a) to35(c), involving a λ/4 MS/MS coupler and a 3 unit cell long MS/LHMcoupled-line coupler, both of approximately equal lengths, equal spacingand propagation constants (see FIG. 32(b)). The return loss (S₁₁),isolation (S₄₁) and coupled power (S₂₁) for each coupler were measuredusing a HP8753D network analyzer and values obtained were compared tomicrowave-circuit simulation results. As shown in FIG. 35(a), thecoupled power of the MS/LHM coupled-line coupler is better than that ofthe MS/MS coupler. Furthermore, from FIG. 35(b), it is seen that theamount of power leaking into the isolated port of the MS/LHMcoupled-line coupler is lower than that of the corresponding MS/MScoupler. In addition, at the design frequency, the return loss of theMS/LHM coupled-line coupler is lower than its conventional counterpart(see FIG. 35(c)). Hence, the metamaterial MS/LHM coupled-line couplerexhibits superior performance in terms of the coupled power, isolationand return loss when compared to a conventional MS/MS coupler of thesame length, line separation and propagation constant.

[0166] As expected from the dispersion analysis and the theory presentedearlier, the MS/LHM coupled-line coupler has a very large coherencelength. Under such conditions, arbitrary coupling can be achieved asdemonstrated in FIG. 35(d). As the number of unit cells is increased,the coupled power improves and it is possible to achieve close to 0 dBcoupled power.

[0167] Overall, there is good agreement between theory, simulation andexperiment. The predictions of the coherence length as well as the levelof the coupled power agree well with the experimentally observedresults. The coupled-line coupler demonstrates better performance interms of coupling, isolation and reflection when compared to aconventional microstrip coupler of the same length, line spacing andpropagation constant.

[0168] Small discrepancies between simulation and experimental resultscan be attributed to a number of factors. The absence of accurate modelsfor the inductors used and their self-resonance characteristics affectthe design. Moreover chip components with finite dimensions add to theoverall electrical length of the MS/LHM coupler which needs to beproperly accounted for.

[0169] In conventional microstrip branch-line couplers, power splitsequally between the through and coupled ports and the ports are −90° and−180° out of phase respectively with reference to the input port.Interesting phase compensation effects can be achieved when some of thebranches of the branch-line couplers are replaced by LHM lines.

[0170] Two different branch-line coupler designs are presented belowdenoted as type 1 and type 2. The type 1 branch-line coupler usesregular microstrip lines for the low-impedance branches and LHM linesfor the high-impedance branches (see FIG. 36(a)). The type 2 branch-linecoupler is the dual of the type 1 branch-line coupler and utilizes LHMlines for the low-impedance branches and microstrip lines for thehigh-impedance branches (see FIG. 36(b)). For both branch-line couplers,power splits equally between the two output ports with a 0°-phase shiftwith respect to the input at the coupled port. Furthermore, the type 1branch-line coupler offers a negative phase quadrature (−90°) while thetype 2 branch-line coupler provides a positive phase quadrature (+90°)at the through port with respect to the input port.

[0171]FIG. 36(a) shows the structure of the type 1 branch-line couplermade of alternating MS and LHM lines. The scattering parameters of thecoupler can be determined using even-odd mode analysis. It can be easilyshown that the impedance looking into a segment of LHM transmission lineof length d is given by: $\begin{matrix}{Z_{i\quad n} = {Z_{0}\frac{Z_{L} + {j\quad Z_{0}\tan \quad {\beta d}}}{Z_{0} + {j\quad Z_{L}\tan \quad \beta \quad d}}}} & (31)\end{matrix}$

[0172] where:

[0173] Z_(L) is the load impedance at the end of the line;

[0174] Z₀ is its characteristic impedance; and

[0175] β is a negative number corresponding to the propagation constant.

[0176] For the purpose of analysis, port 1 is excited by a source V_(s)of internal impedance Z₀ and all other ports are terminated at Z₀. Inthe even mode analysis, both port 1 and port 2 are excited by V_(s)/2sources. In the odd mode, the source at port 1 is V_(s)/2 and the sourceat port 2 is −V_(s)/2 (see FIG. 37).

[0177] In the even mode, looking down the LHM lines at port 1 and port 3(see FIG. 36(a)), one sees open λ/8 lines and hence βd is −π/4. Fromequation (31), this corresponds to shunt loading of the λ/4 MS line atits ends by a susceptance −jY₀ (see FIG. 37). In the odd mode, one canobserve shorted λ/8 segments of LHM lines instead, corresponding toshunt loading of the MS line by a susceptance jY₀.

[0178] By analyzing the equivalent circuits in FIG. 37, it is seen thatthe impedance looking into each port is Z₀. Since the ports are alwaysmatched, S₁₁, S₂₂, S₃₃ and S₄₄ are zero. As the waves in each porttravel in only one direction, by superposition of the even and oddmodes, S₂₁ is zero. In the even mode, the signal at port 1 can berelated to the signal at port 3 by the following transfer matrices:$\begin{matrix}{\begin{bmatrix}{V_{s}/4} \\I_{1}\end{bmatrix} = {{{\begin{bmatrix}1 & 0 \\{{- j}\quad Y_{0}} & 1\end{bmatrix}\begin{bmatrix}0 & {j\frac{\quad Z_{0}}{\sqrt{2}}} \\{j\quad Y_{0}\sqrt{2}} & 0\end{bmatrix}}\begin{bmatrix}1 & 0 \\{{- j}\quad Y_{0}} & 1\end{bmatrix}}\begin{bmatrix}V_{3} \\{V_{3}Y_{0}}\end{bmatrix}}} & (32) \\{{V_{3}^{even}} = {{\frac{1}{\sqrt{2}}\frac{V_{s}}{4}( {1 - j} )} = V_{4}^{even}}} & (33)\end{matrix}$

[0179] In the even mode, the voltage at port 3 and port 4 are identical(see FIG. 37). In the odd mode, the transfer matrices are:$\begin{matrix}{\begin{bmatrix}{V_{s}/4} \\I_{1}\end{bmatrix} = {{{\begin{bmatrix}1 & 0 \\{j\quad Y_{0}} & 1\end{bmatrix}\begin{bmatrix}0 & {j\frac{Z_{0}}{\sqrt{2}}} \\{j\quad Y_{0}\sqrt{2}} & 0\end{bmatrix}}\begin{bmatrix}1 & 0 \\{j\quad Y_{0}} & 1\end{bmatrix}}\begin{bmatrix}V_{3} \\{V_{3}Y_{0}}\end{bmatrix}}} & (34) \\{\begin{bmatrix}{{- V_{s}}/4} \\I_{2}\end{bmatrix} = {{{\begin{bmatrix}1 & 0 \\{j\quad Y_{0}} & 1\end{bmatrix}\begin{bmatrix}0 & {j\frac{Z_{0}}{\sqrt{2}}} \\{j\quad Y_{0}\sqrt{2}} & 0\end{bmatrix}}\begin{bmatrix}1 & 0 \\{j\quad Y_{0}} & 1\end{bmatrix}}\begin{bmatrix}V_{4} \\{V_{4}Y_{0}}\end{bmatrix}}} & (35) \\{{ \Leftrightarrow V_{3}^{odd}  = {{- \frac{1}{\sqrt{2}}}\frac{V_{s}}{4}( {1 + j} )}};{V_{4}^{odd} = {\frac{1}{\sqrt{2}}\frac{V_{s}}{4}( {1 + j} )}}} & (36)\end{matrix}$

[0180] where equation (34) is the transfer matrix from port 1 to port 3and equation (35) is the transfer matrix from port 2 to port 4 (see FIG.37). The amplitude of the total incident voltage (odd and even) at port1 is V_(s)/2. Hence from superposition of the modes, S₃₁ and S₄₁ areequal to −j/{square root}2 and 1/{square root}2 respectively. Thescattering matrix for the MS/LHM branch-line coupler is: $\begin{matrix}{\begin{bmatrix}V_{1}^{-} \\V_{2}^{-} \\V_{3}^{-} \\V_{4}^{-}\end{bmatrix} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 0 & {- j} & 1 \\0 & 0 & 1 & {- j} \\{- j} & 1 & 0 & 0 \\1 & {- j} & 0 & 0\end{bmatrix}}\begin{bmatrix}V_{1}^{+} \\V_{2}^{+} \\V_{3}^{+} \\V_{4}^{+}\end{bmatrix}}} & (37)\end{matrix}$

[0181] From equation (37), it is seen that in the MS/LHM branch-linecoupler, the power divides equally between the through port (port 3) andcoupled port (port 4) and they are 90° and 0° out of phase respectivelywith reference to the input (port 1).

[0182] To construct the type 1 branch-line coupler a substrate similarto that of the coupled-line couplers was used. In the type 1 branch-linecoupler, the MS strips have an impedance of 50/{square root}2Ω whereasLHM strips have an impedance of 50 Ω (using a microwave-circuitsimulator). The design frequency is chosen to be 1.9 GHz so that eachside of the coupler is λ/4 long (from Matlab® simulations of thedispersion characteristics of the line). The MS line used is 4.018 mmwide and 20.603 mm long. The LHM-line strip comprises 3 unit cells ofwidth 2.343 mm and length 5.2 mm each. The latter is loaded with 3.3 nHshunt inductors and 1.6 pF series capacitors. The end capacitors of eachLHM line segment are 2.4 pF. The terminating capacitors are chosen to bealmost twice the value of the interior ones to preserve the symmetry ofthe unit cells making up the LHM lines (see FIG. 31(b)).

[0183] During simulation, port 1 of the branch-line coupler shown inFIG. 36(a) is excited in a microwave-circuit simulation and all otherports are terminated with 50Ω loads. The reflected power (S₁₁),isolation (S₂₁), through power (S₃₁) and coupled power (S₄₁) arerecorded and compared to theoretical estimates from even-odd modeanalysis. To apply the previously presented even-odd mode analysis,Matlab® simulations are performed for both lines to determine theirdispersion characteristic. Subsequently, the corresponding phase shiftsare used in the even-odd mode analysis. These theoretical results arecompared with a microwave-circuit simulation and are presented in FIGS.38(a) to 38(c).

[0184] In this branch-line coupler, the input power splits in halfbetween the through port (port3) and coupled port (port 4) as expected(see equation (22)). This is demonstrated in FIGS. 38(a) and 38(b).FIGS. 38a and 38 b also show that at the design frequency, the throughand coupled ports are matched (see S₁₁ in FIG. 38(a)) and port 2 isisolated (see FIG. 38(b)). In FIGS. 38(a) and 38(b) the response of acommensurate, conventional, branch-line coupler is also included forcomparison purposes. As shown, there is no degradation of the usefulbandwidth between the conventional and the type 1 branch-line coupler.However, from theory the through port (port 3) undergoes a phase shiftof −90° and the coupled port (port 4) undergoes a phase shift of 0° withrespect to the input. This is clearly demonstrated from the simulationresults in FIG. 38(c).

[0185] The schematic for the type 2 MS/LHM branch-line coupler is shownin FIG. 36(b). The scattering parameters for this branch-line couplercan be determined in the same way followed for the type 1 case. Since inthis branch-line coupler, the MS and LHM lines are switched, everyoccurrence of j in equations (32) to (37) is replaced by −j. Hence thescattering matrix for the type 2 MS/LHM branch-line coupler is:$\begin{matrix}{\begin{bmatrix}V_{1}^{-} \\V_{2}^{-} \\V_{3}^{-} \\V_{4}^{-}\end{bmatrix} = {{\frac{1}{\sqrt{2}}\begin{bmatrix}0 & 0 & j & 1 \\0 & 0 & 1 & j \\j & 1 & 0 & 0 \\1 & j & 0 & 0\end{bmatrix}}\begin{bmatrix}V_{1}^{+} \\V_{2}^{+} \\V_{3}^{+} \\V_{4}^{+}\end{bmatrix}}} & (38)\end{matrix}$

[0186] From equation (38), it is seen that in the type 2 MS/LHMbranch-line coupler, the power divides equally between the through port(port 3) and coupled port (port 4) and the parts are +90° and 0° out ofphase respectively with reference to the input (port 1).

[0187] For the type 2 branch-line coupler, the LHM strips are designedwith 50/{square root}2 Ω impedance and the MS strips with 50 Ωimpedance. The design frequency is chosen at 1.7 GHz so that again allsides are λ/4 long. In this case, the MS lines used are 2.342 mm wideand 23.645 mm long. The LHM-line strip comprises 3 unit cells of width4.017 mm and length 5 mm each. The latter is loaded with 2.7 nH shuntinductors and 2.4 pF series capacitors, with terminating 4 pF capacitorsfor the LHM lines.

[0188] The type 2 branch-line coupler splits the input power equallybetween the two forward ports (through and coupled) and the backwardport is isolated. As will be appreciated, the magnitude response is verysimilar to that of the type 1 branch-line coupler. From theory (equation(23)) and verified from the simulation results in FIG. 38(d), thethrough port (port 3) undergoes a phase shift of 90° and the coupledport (port 4) undergoes a phase shift of 0° with respect to the input.

[0189] A conventional branch-line coupler allows for equal power splitbetween the output ports but with −90° and −180° phase shifts withrespect to the input (see FIGS. 38(a) to 38(c)). As will be appreciated,the present branch-line couplers offer more options in the design ofcircuits involving branch-line couplers. than available withconventional branch-line couplers. Specifically either a positive or anegative phase quadrature at the through port can be achieved,associated with a phase compensation at the coupled port.

[0190] The present coupled-line coupler demonstrates superiorperformance in terms of coupling, isolation and return loss whencompared to a conventional microstrip coupled-line coupler of the samelength, line spacing and propagation constant. Moreover, it is possibleto realize such MS/LHM coupled-line couplers to achieve an arbitrarycoupling coefficient. Furthermore, MS/LHM branch-line couplers possesssimilar functionality to their conventional counterparts whenconsidering the magnitude of the power transfer between the ports.However, new functionality is added by the ability to obtain phasecompensation at the coupled port (0° phase shift with respect to theinput) along with a choice of either a positive or a negative phasequadrature (±90° phase shift) at the through port. On the other hand,the corresponding bandwidth remains comparable to that of conventional,commensurate, microstrip branch-line couplers.

[0191] Backward Leaky-Wave Antenna Using Metamaterial

[0192] Charged particles traveling at speeds greater than the phasevelocity of light in a medium emit coherent radiation better known asCherenkov radiation. The angle of the radiated conical wave front isgiven by the velocity of the particle (V) with respect to the phasevelocity of EM waves (v) within the medium in the following manner:$\begin{matrix}{{{Cos}(\theta)} = {{v/V} = \frac{c/n_{0}}{V}}} & (39)\end{matrix}$

[0193] where:

[0194] n_(o) is the refractive index of the surrounding medium;

[0195] c is the speed of light in a vacuum; and

[0196] θ is the angle between the particle velocity and the radiated EMwave front. This expression suggests that in a medium with negative no,the angle θ becomes obtuse. This implies that radiation is directedbackward rather than forward, as is the case in a RHM.

[0197] Similarly, if a periodic guiding structure supports Blochcurrents (moving charge) or equivalently EM waves with phase velocitiesgreater than the speed of light, the angle of the radiated wave front isderived from equation (39) by letting V be the phase velocity of the EMwave along the guiding interface. This is due to the phase matchingcondition along the interface of the guide and surrounding medium. Ifthe guiding structure is a medium with an effective refractive index n₁,and the surrounding medium has a refractive index n₀, then the angle ofthe radiated wave front is given by the following expression:$\begin{matrix}{{{Cos}(\theta)} = {\frac{c/n_{0}}{c/n_{1}} = \frac{n_{1}}{n_{0}}}} & (40)\end{matrix}$

[0198] The above equation indicates that if the refractive index of theguiding medium (n₁) is negative, the radiation emitted into a RHM willbe directed backward. It is in fact the phase advancement (backward-wavepropagation) that causes backward radiation into the surrounding medium.

[0199] In order to excite backward-wave radiation from a dualtransmission line LHM into free space, the L, C parameters need to bechosen such that the effective refractive index of the LHM, (n₁), isnegative and smaller in magnitude than one, as indicated by equation(40). Equivalently, the magnitude of the propagation constant along theguiding LHM needs to be designed to be smaller in magnitude than thepropagation constant in free-space k_(o).

[0200] A radiating LHM antenna at 15 GHz based on dual transmissionlines of the type shown in FIG. 13 was created. The LHM antenna includes16 unit cells, with each unit cell having a dimension of 4.268 mm,approximately a factor of 5 smaller than the free-space wavelength of 2cm at 15 GHz. Thus, the antenna can be treated as an effective medium.Commercial method of moments software was used to design the layout ofthe antenna. The LHM was fabricated by a mask/photo-etching technique ona 20-mil thick Rogers™ RO3203 circuit board with a relative permittivityof 3. The interconnecting transmission line sections were implemented incoplanar waveguide (CPW) technology. The transmission lines include aplanar center conductor with two adjacent ground planes on either side.In the coplanar waveguide configuration, both ground planes and thecenter conductor lie in the same plane, which allows the simpleintegration of shunt inductors and series capacitors. The gaps in theCPW center conductor serve as the series capacitors and the narrow linesconnecting the center conductor to the coplanar ground planes act as theshunt inductors. It is in fact these capacitive gaps that radiate in thestructure and cause a backward emerging transverse magnetic wavefront.On the contrary, the inductive lines are non-radiating due to theantiparallel currents flowing on each pair of inductive lines. This oddsymmetry causes cancellation in the far field and leads to lowcross-polarization levels.

[0201] Although there are various radiating structures that exhibitphase and group velocities of opposite sign, the proposed structure,unlike conventional structures operates in the long wavelength regimeand demonstrates backward-wave radiation in its lowest passband ofoperation. The proposed structure supports a backward-wave fundamentalspatial harmonic that radiates. Early examples of backward-waveradiating structures include the helix antenna, corrugated dielectric,or metallic surfaces, and periodic arrays of radiating elements fed byslow wave transmission line sections of large periodic spacing (d>λ/2).These structures, however, radiate in higher-order passbands (exhibithigher-order radiating spatial harmonics) and therefore, effectivematerial constants such as a refractive index cannot in general bedefined. On the other hand, log-periodic dipole arrays and relateduniform dipole arrays with a transposed feed have shown to producebackward waves even for small longitudinal periodic spacings.Nevertheless, the dipoles are resonant so element dimensions stillremain electrically large (˜λ/2) and therefore effective materialparameters, e and m cannot be defined.

[0202] Super-Lens Formed From Metamaterials and RHM

[0203] The present metamaterial is also suitable for use in a focusingdevice that overcomes diffraction limits, hereinafter referred to as a“super-lens”. The super-lens includes two-field focusing devices of thetype previously discussed and is formed by sandwiching the LHM betweenthe two RHM at two parallel interfaces. The LHM is a planar slab havinga grid of printed metallic strips over a ground plane, loaded withseries capacitors C and shunt inductors L. The two RHMs are two unloadedprinted grids which act as homogeneous and isotropic media with positiveindices of refraction.

[0204]FIG. 39 illustrates the planar transmission line left-handedsuper-lens. The unit cell of the left-handed grid is shown in the topinsert while the unit cell of the positive refractive index grid isshown in the bottom insert. The super-lens is fabricated on a groundedmicrowave substrate (Rogers™ RO3003) of thickness 60 mil (1.52 mm) anddielectric constant ε_(r)=3.00. The left-handed slab has a 5×19 grid ofprinted metallic strips (microstrip lines) loaded with series chipcapacitors C and shunt chip inductors L to the ground. Each unit cellhas dimensions 8.40 mm×8.40 mm and therefore is much smaller than thenominal operating wavelength of 15.59 cm at 1 GHz. Hence, the loadedperiodic grid acts as an effective medium exhibiting a negative index ofrefraction. The left-handed planar slab is sandwiched between twocommensurate unloaded printed grids that act as effective homogenousmedia with positive indices of refraction. The unloaded grid and groundplane behave like a dielectrically loaded parallel-plate waveguide, butin addition allow simple measurement of the guided fields throughproximity coupling. The left-handed grid is designed so that it isimpedance matched and exhibits a refractive index of −1 relative to theunloaded grids at the design frequency of 1.00 GHz.

[0205] The first unloaded grid is excited at a point which is imaged bythe left-handed slab to the second unloaded grid, with a verticalmonopole fed by a coaxial cable through the ground plane. The monopoleattaches the center conductor of the coaxial cable to a point on theunloaded grid, while the outer conductor of the coaxial cable attachesto the ground plane. The vertical monopole lies along the center row(row 0), located 2.5 unit cells away from the first interface of theleft-handed slab. According to geometrical optics, the source and imageshould be symmetrically positioned with respect to the left-handed slab.This is due to the fact that the distance from the source to the firstinterface (2.5 unit cells) is half the left-handed slab thickness (5cells). The set-up shown in FIG. 39 is a practical implementation forimaging a point source from one homogeneous dielectric to another usinga left-handed slab. The vertical electric field over the entirestructure is measured using a detecting probe. The field is detected 0.8mm above the entire surface of the structure, using a short verticalprobe connected to a Hewlett-Packard Vector Network Analyzer model8753D. Port 1 of the network analyzer is connected to the coaxial cablethat feeds the exciting monopole. Port 2 is connected by a separatecoaxial cable to the detecting probe that is scanned above the surfaceof the structure using a computer-controlled stepper motor. The measuredtransmission coefficient is proportional to the voltage of the gridnodes with respect to the ground plane. The best focusing results wereobserved at 1.057 GHz, a frequency slightly higher than the designfrequency of 1.00 GHz. This is primarily due to the variation of chipinductors and capacitors from their nominal values, as well asfabrication tolerances in printing the grid lines. The measured verticalelectric field above each unit cell for the entire structure is shown inFIG. 40, at a frequency of 1.057 GHz. The plot is normalized withrespect to the source amplitude. The source is located at (column,row)=(0,0) and the image at (10,0), whereas the first and secondinterfaces are located between columns 2 & 3 and 7 & 8, respectively. Asshown, the enhancement of evanescent waves is quite evident at thesecond interface of the left-handed slab near the center row (row 0) andthe results agree well with microwave-circuit simulations. FIG. 41explicitly shows the measured electric field along row 0 to emphasizethe growing evanescent fields within the left-handed lens. The verticaldashed lines identify the source at column 0 and the image at column 10while the vertical solid lines identify the interfaces of theleft-handed slab.

[0206] Geometrical optics establish that the source and its image shouldbe separated by twice the thickness of the slab (2×5=10 unit cells).Thus, the measured vertical electric field along the image column(column 10) is shown in FIG. 42. Plotted in the same Figure are themeasured transverse patterns at the source column (column 0), as well asthe theoretical diffraction-limited pattern. All patterns in FIG. 42 arenormalized to unity for comparison purposes. Nevertheless, the measuredsource and image peaks lie within 7% of each other. In addition, it isimportant to explain how the theoretical diffraction-limited pattern wascomputed. Due to the close proximity (0.54 effective wavelengths) of thesource and its image, evanescent waves that reach the image should beaccounted for. For this reason, the diffraction-limited pattern assumesthat the propagating components are focused whereas the evanescentcomponents are not neglected, but rather assumed to exponentially decayfrom the source to the image, with attenuation factors corresponding toa refractive index of n=+1. Expressions for the voltage (V) at the imageplane (column 10) are given by the integral in equation 41 below:$\begin{matrix}{{V(x)} = {C{\int_{- \infty}^{\infty}{\frac{^{\quad k_{z1}D}^{\quad k_{z2}D}^{\quad k_{x}x}}{k_{z1}}{k_{x}}}}}} & (41)\end{matrix}$

 k _(z1) =−k _(z2) ={square root}{square root over (k_(o) ²−k_(x) ²)}for k _(.x.) <k _(o) (propagating waves)

k _(z1) =k _(z2) =i{square root}{square root over (k_(x) ²−k_(o) ²)} fork _(.x.) >k _(o) (evanescent waves)

[0207] where:

[0208] C is a constant that normalizes V(0)=1;

[0209] D is the width of the lens; and

[0210] k_(o) is the intrinsic wavenumber in all grid media.

[0211] The half-power beamwidth of the diffraction-limited image shownin FIG. 42 is 129 degrees. It was found that the diffraction-limitedimage including attenuating evanescent waves and the image neglectingthem altogether are almost identical.

[0212] With this clarification in mind, a comparison of the images shownin FIG. 42 can be made. Clearly, the measured image pattern is narrowerthan the theoretical diffraction-limited one. The measured half-powerbeamwidth is 75 degrees. This is significantly narrower than the 129degrees for the diffraction-limited image. This establishes that it ispossible to overcome the diffraction limit when imaging from onehomogeneous dielectric to another through a left-handed isotropic lens.This narrowing of the beamwidth beyond the diffraction limit can beascribed to the enhancement of the evanescent waves evident in FIGS. 39and 40. Nevertheless, the image is still imperfect since the sourcebeamwidth is narrower than that of the image. This is not surprisingconsidering that slight material losses and mismatches at the lensinterfaces lead to departures from the n=−1 condition and thus, degradethe ability of a left-handed lens to achieve perfect imaging.

[0213] Resonance Cone Antenna

[0214] Resonance cone metamaterial may also be used in an antenna, byshunting the metamaterial with inductors, at the transmission lineintersections. FIG. 43 shows a resonance cone antenna 4300 constructedusing metamaterial. The antenna includes a 2×3 array of macrocells. Eachmacrocell includes a plurality of microcells of orthogonal pairs oftransmission line segments joined in a substantially rectangularfashion. The segments are loaded periodically with lumped capacitors inone direction and lumped inductors in a perpendicular direction. Themicrocells of the macrocells have a shared orientation that is differentfrom the orientation of microcells in adjacent macrocells by 90 degrees.Macrocells are joined by transition regions similar to those describedpreviously with reference to the focusing device. The transmission linesegments of the transition regions include both lumped capacitors andinductors arranged in a manner consistent with FIG. 24. The deviceinput, or feed point, is located at the transition region adjacent thecenter of the macrocell array.

[0215] This arrangement of metamaterials, and input location allowsinput waves of compatible wavelength, to form a resonance cone patern oftwo squares that are multi-refracted within the macrocell array. Thisgives radiation patterns that can be close to the ground plane orelevated, but do not suffer from unwanted edge interactions as shown inFIGS. 44 and 45. Moreover, as the frequency departs from the designvalue, the resonance cone squares turn into expanding or contractingsquare spirals that become weak before reaching the grid edges due toradiation or ohmic losses. This phenomenon is inherently non-resonantwhich leads to broadened operating bandwidths as compares to patchantennas over ground.

[0216] As will be appreciated it has been shown that negative refractionand focusing of electromagnetic waves can be achieved in metamaterialsthat support backward waves without employing resonances or directlysynthesizing the permittivity and permeability. Schemes for fabricatingsuch media by appropriately loading a host transmission line medium havealso been described. The resulting planar topology permits LHMstructures to be readily integrated with conventional planar microwavecircuits and devices and used in a variety of applications to guide andcontrol electromagnetic radiation.

[0217] Although preferred embodiments of the present invention have beendescribed, those of skill in the art will appreciate that variations andmodifications may be made without departing from the spirit and scopethereof as defined by the appended claims.

What is claimed is:
 1. A planar metamaterial comprising: twosubstantially orthogonal, coplanar sets of transmission lines, saidtransmission lines being spaced with a periodicity, loaded withcapacitors with said periodicity, and shunted with inductors with saidperiodicity such that for an electromagnetic wave, having a wavelengthgreater than said periodicity and traveling along the plane of saidtransmission lines, the effective permeability and permittivity of saidmetamaterial are simultaneously negative.
 2. A planar metamaterialaccording to claim 1 wherein the wavelength of said electromagnetic waveis greater than said peridicity by an order of magnitude.
 3. A linearmetamaterial comprising: a transmission line, having a linear dimension,and being loaded with capacitors, and shunted with an inductor such thatfor an electromagnetic wave, having a wavelength greater than saidlinear dimension and traveling along the axis of said transmission line,the effective permeability and permittivity of said metamaterial aresimultaneously negative.
 4. A planar metamaterial according to claim 3wherein the wavelength of said electromagnetic wave is greater than saidperiodicity by an order of magnitude.
 5. A metamaterial according toclaim 4 wherein the values of said capacitors and inductor areproportioned to provide a continuous passband for said electromagneticwave.
 6. A metamaterial according to claim 4 wherein said transmissionline includes two substantially identical capacitors on either side ofsaid inductor.
 7. A metamaterial according to claim 4 further comprisinga plurality of substantially identical transmission lines arranged inseries.
 8. A metamaterial according to claim 7 wherein each transmissionline includes two substantially identical capacitors on either side ofsaid inductor.
 9. A metamaterial according to claim 8 wherein the valuesof said capacitors and inductor are proportioned to provide a continuouspassband for said electromagnetic wave.
 10. A planar resonance conemetamaterial comprising: a first set of transmission lines, spaced witha periodicity, and loaded with capacitors with said periodicity; asecond set of transmission lines, substantially orthogonal and coplanarwith said first set of transmission lines, said second set oftransmission lines being spaced with said periodicity, and loaded withinductors with said periodicity, said first and second sets oftransmission lines exhibiting characteristics such that for anelectromagnetic wave, having a wavelength greater than said periodicityby an order of magnitude, and traveling along the linear axis of saidfirst set of transmission lines, the effective permittivity of saidmetamaterial is positive such that for an electromagnetic wave, having awavelength greater than said periodicity by an order of magnitude andtraveling along the linear axis of said second set of transmissionlines, the effective permittivity of said metamaterial is negative. 11.A metamaterial according to claim 10 further comprising an input on theperimeter of said metamaterial coincident with an intersection of onetransmission line of said first set and one transmission line of saidsecond set and a plurality of outputs on the perimeter of saidmetamaterial coincident with intersections of transmission lines of saidfirst and second sets.
 12. A planar resonance cone metamaterialcomprising: a first set of transmission lines, spaced with a firstperiodicity, and loaded with capacitors with a second periodicity; asecond set of transmission lines, substantially orthogonal and coplanarwith said first set of transmission lines, said second set oftransmission lines being spaced with said second periodicity, and beingloaded with inductors with said first periodicity whereby for anelectromagnetic wave, having a wavelength greater than said first andsecond periodicities by an order of magnitude and traveling along thelongitudinal axis of said first set of transmission lines, the effectivepermittivity of said metamaterial is positive and for an electromagneticwave having a wavelength greater than said first and second priodocitiesand traveling along the longitudinal axis of said second set oftransmission lines, the effective permittivity of said metamaterial isnegative.
 13. A near field focusing device comprising: a first set oftransmission lines, said first set of transmission lines being spacedwith a first periodicity and loaded with capacitors with a secondperiodicity; a second set of transmission lines, substantiallyorthogonal to, and coplanar with said first set of transmission lines,said second set of transmission lines being spaced with said secondperiodicity, loaded with capacitors with said first periodicity, andshunted with inductors, said first set of transmission linesintersecting said second set of transmission lines such that for anelectromagnetic wave, having a wavelength greater than said first orsecond periodicity by an order of magnitude, and traveling along theplane of said transmission lines, the effective permeability andpermittivity of said metamaterial are simultaneously negative; and aplanar waveguide, having a flat extent, coupled to said transmissionlines, such that said flat extent is parallel to one set of transmissionlines.
 14. A phase-shifting device comprising: a transmission line,having a linear dimension and characteristic impedance; capacitorsloaded on said transmission line; and an inductor shunting saidtransmission line, said transmission line exhibiting characteristicssuch that for an electromagnetic wave, having a wavelength greater thansaid linear dimension by an order of magnitude and traveling along theaxis of said transmission line, the effective permeability andpermittivity of transmission are simultaneously negative and such thatsaid transmission line provides a continuous passband for saidelectromagnetic wave along said transmission line.
 15. A phase-shiftingdevice according to claim 14 further comprising a plurality oftransmission lines coupled in series.
 16. A phase-shifting-deviceaccording to claim 15 wherein each transmission line includes a pair ofidentical capacitors on either side of said inductor.
 17. A near fieldfocussing device comprising: a parallel-plate waveguide; and a negativerefractive index metamaterial in line with said waveguide and forming aninterface therewith.
 18. A focussing device according to claim 17wherein said waveguide serves as a right-handed medium, electromagneticradiation passing through said interface from said waveguide beingfocussed in said metamaterial.
 19. A coupled-line coupler comprising: amicrostrip line; and a left-handed material coupled to an edge of saidmicrostrip line.
 20. A coupled-line coupler according to claim 19wherein said left-handed material is synthesized by loading a microstriptransmission line with series capacitors and shunt inductors.
 21. Acoupled-line coupler according to claim 20 including an input port andthrough and coupled ports thereby to define a branch-line coupler.